The non-dimensional governing equations (REF ), boundary conditions (REF )–(REF ) and flux constraints (REF )–(REF ), define a 3D boundary-value problem for \(u\) , \(p\) , \(c\)  and \(\Gamma \) .
The solution depends on 9 dimensionless groups (\(\epsilon \) , \(\) , \(\) , \(_I\) , \(\) , \(\) , \(\phi _x\) , \(\phi _z\)  and \(P_z\) ) that characterise the geometry, flow, liquid and surfactant.
The number of dimensionless groups differs from [1]} (where there are 8), as the authors considered 2D geometries, and [2]} (where there are 10), as we have absorbed the non-dimensional background concentration into \(\) , similar to [1]}.
Our aim is to construct a reduced model in the limit \(\epsilon \ll 1\)  to predict the drag reduction (REF ).
